https://file.aiquickdraw.com/mathbot/file/1729294514516rtm1a9o1.jpg

Let's break down the given function \( f(x) = 3[-2(x + 4)]^2 - 1 \) and answer the questions on the chart.

1. **Parent Graph:**
   - The parent function is \( f(x) = x^2 \), which is a standard quadratic function (a parabola).

2. **Vertical Translation:**
   - The function has a vertical shift of \(-1\), which means the graph moves *down* by 1 unit.

3. **Horizontal Translation:**
   - Inside the function, we have \( (x + 4) \). This represents a horizontal shift to the left by 4 units.

4. **Stretch/Compress (Vertical):**
   - The coefficient of 3 outside the squared term indicates a vertical stretch by a factor of 3.

5. **Stretch/Compress (Horizontal):**
   - The factor \(-2\) inside the squared term represents a horizontal compression by a factor of \( \frac{1}{2} \) (since \( |2| \) is the horizontal compression factor).

6. **Flipped?**
   - The negative sign inside the squared term indicates a reflection across the y-axis (upside down along the horizontal axis).

### Summary for the chart:

- **Vertical Translation:** The graph moved *down* 1 unit.
- **Horizontal Translation:** The graph moved *left* 4 units.
- **Vertical Stretch:** The graph is *stretched* vertically by a factor of 3.
- **Horizontal Compression:** The graph is *compressed* horizontally by a factor of 2.
- **Flipped:** The graph is reflected (flipped upside down).

Would you like more details or clarification on any of the transformations?

Here are 5 related questions:
1. How would the graph change if the coefficient outside was negative, like \( -3 \)?
2. What would happen if the horizontal shift was \( (x - 4) \) instead of \( (x + 4) \)?
3. How do you identify vertical versus horizontal stretches or compressions in a function?
4. How can you determine the direction of a graph’s flip based on its equation?
5. What does the general form \( a(x - h)^2 + k \) tell you about transformations?

**Tip:** Always check both inside and outside the function to identify horizontal and vertical transformations separately.

Given the function f(x) = 3[-2(x + 4)]^2 - 1, determine the vertical translation, horizontal translation, stretch/compression, and reflection.

Explore the transformations of the quadratic function f(x) = 3[-2(x + 4)]^2 - 1, including vertical and horizontal shifts, stretches/compressions, and reflections. Understand how each component affects the graph's shape and position.

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